Question

A body moves in a circular orbit of radius R under the action of a central force. Potential due to the central force is given by ${V_r} = kr$ (k is a positive constant). Period of revolution of the body is proportional to
A. ${R^{\dfrac{1}{2}}}$
B. ${R^{\dfrac{{ - 1}}{2}}}$
C. ${R^{\dfrac{{ - 3}}{2}}}$
D. ${R^{\dfrac{{ - 5}}{2}}}$

Answer 1

Hint: Central force is a conservative force which is expressed as $F = \dfrac{{d{V_r}}}{{dr}}$, where ${V_r}$ is the potential energy. Find the value of central force by substituting the value of potential energy. This central force balances the centripetal force acting on the body revolving in a circular orbit of radius R, which means $F = \dfrac{{m{v^2}}}{R}$. Equate the obtained central force with the centripetal force to find the value of the velocity. Substitute the value of velocity in the formula of time period of revolution $T = \dfrac{{2\pi R}}{v}$ and find the value of the time period.

Complete step by step answer:
We are given that a body moves in a circular orbit of radius R under the action of a central force and has Potential energy due to the central force as ${V_r} = kr$ (k is a positive constant).
Find the central force by substituting the value of potential energy.
$
  F = \dfrac{{d{V_r}}}{{dr}} \\
  {V_r} = kr \\
  F = \dfrac{{d\left( {kr} \right)}}{{dr}} \\
  d\left( {kr} \right) = kdr \\
  F = k\dfrac{{dr}}{{dr}} \\
  F = k \\
 $
 This central force balances the centripetal force acting on the body revolving in a circular orbit of radius R.
Which means the central force and the centripetal force are equal.
 $
  F = \dfrac{{m{v^2}}}{R} \\
  F = k \\
  \Rightarrow k = \dfrac{{m{v^2}}}{R} \\
  \Rightarrow {v^2} = \dfrac{{kR}}{m} \\
  \Rightarrow v = \sqrt {\dfrac{{kR}}{m}} \\
 $
Period of revolution of the body is given by $T = \dfrac{{2\pi R}}{v}$
$
  T = \dfrac{{2\pi R}}{v} \\
  v = \sqrt {\dfrac{{kR}}{m}} \\
  \Rightarrow T = \dfrac{{2\pi R}}{{\sqrt {\dfrac{{kR}}{m}} }} \\
  \Rightarrow T = \dfrac{{2\pi R\sqrt m }}{{\sqrt {kR} }} \\
  \Rightarrow T = \dfrac{{2\pi \sqrt {mR} }}{{\sqrt k }} \\
  \Rightarrow T = 2\pi \sqrt {\dfrac{{mR}}{k}} \\
 $
This shows that Time period of the body is directly proportional to ${R^{\dfrac{1}{2}}}$
$T\propto {R^{\dfrac{1}{2}}}$
The correct option is Option A.

Note:Centripetal force is defined as the force that is necessary to keep an object moving in a curved path and that is directed inward toward the center of rotation while centrifugal force is defined as the force that is felt by an object moving in a curved path that acts outwardly away from the center of rotation. So, do not confuse centripetal force with centrifugal force.